Proving Program Correctness: Axiomatic Approach to Partial and Total Correctness

Proving Program Correctness The Axiomatic Approach

What is Correctness? • Correctness: • partial correctness + termination • Partial correctness: • Program implements its specification

Proving Partial Correctness • Goal: prove that program is partially correct • Approach: model computation with predicates • Predicates are boolean functions over program state • Simple example • {odd(x)} a = x {odd(a)} • Generally: {P} S {Q}, where • P  precondition • Q  postcondition • S  Programming language statement

Proof System • Two elements of proof system • Axioms: capture the effect of prog. lang. stmts. • Inference rules: compose axioms to build up proofs of entire program behavior • Let’s start by discussing inference rules and then we’ll return to discussing axioms

Composition • Rule: • Consider two predicates • {odd(x+1)} x = x+1 {odd(x)} • {odd(x)} a = x {odd(a)} • What is the effect of executing both stmts? • {odd(x+1)} x = x+1 ; a = x {odd(a)}

Consequence 1 • Rule • Ex: • {odd(x)} a = x {odd(a)} and • Postcondition  {a  4} • What can we say about this program?

Consequence 2 • Rule: • Ex: • Precondition  {x=1} and • {odd(x)} a = x {odd(a)} • What can we say about this program?

Axioms • Axioms explain the effect of executing a single statement • Axioms will be derived “backwards.” • Start with postcondition and determine what conditions must be true on entry to stmt.

Assignment Axiom • Rule: • Replace all free occurences of x with y • e.g., {odd(x)} a = x {odd(a)}

Conditional Stmt 1 Axiom • Rule: {P} Bif {P  Bif } {P Bif} S {Q}

Example: if even(x) then { x = x +1 } {odd(x)  x > 3} else part (?? even(x)  (odd(x) x>3) then part: {odd(x+1)  x>2} x = x+1 {odd(x)  x > 3} (??  even(x))  (odd(x+1)  x>2) P  ((odd(x+1)  x>2)  x >3) x > 3 works as well. Conditional Stmt 1

Conditional Stmt 2 Axiom • Rule {P} Bif {P  Bif } {P Bif} S1 S2 {Q}

Example: if x < 0 then { x = -x; y = x else y = x } {y = |x|} Then part: {x = |x|} y = x {y = |x|} {-x = |x|} x = -x {x = |x|} ( ??  x <0)  -x = |x| Else part: {x =|x|} y=x{y=|x|} ( ??  ¬(x < 0))  x = |x| P  (-x = |x|)  (x=|x|) Conditional Stmt 2 Axiom

While Loop Axiom • Rule • Infinite number of paths, so we need one predicate for that captures the effect of S • P is called an invariant {P} Bif S {P B}

Example IN  {B  0} a = A b = B y = 0 while b > 0 do { y = y + a b = b - 1 } OUT  {y = AB} INV  y + ab = AB  b  0 Bw  b > 0 Show INV  ¬ Bw  OUT y + ab = AB  b  0  ¬(b > 0) y + ab = AB  b = 0 y = AB So {INV  ¬ Bw}  OUT Establish IN  INV {ab = AB  b  0} y=0 { INV} {aB = AB  B  0} b = B {….} {AB = AB  B  0} a = A {….} So {IN} a=A;b=B;y=0 {INV} While Loop Axiom

While Loop Axiom • Need to show {INV  Bw} loop body {INV} {y+a(b-1) = AB  b-1  0} b = b - 1 {INV} {y+a+a(b-1) = AB  b-1  0} y = y+a {….} {y +ab = AB  b-1  0} loop body {INV} • y + ab = AB  b  0  b > 0 {y +ab = AB  b-1  0}, • So • {IN} lines 1-3} {INV}, • {INV} while loop {INV  ¬ Bw }, and • {INV  ¬ Bw}  OUT • Therefore • {IN} program {OUT}

Total correctness • After you have shown partial correctness • Need to prove that program terminates • Usually a progress argument. Last program • Loop terminates if b  0 • b starts positive and is decremented by 1 every iteration • So loop must eventually terminate

Proving Program Correctness: Axiomatic Approach to Partial and Total Correctness